On Necessary and Sufficient Conditions for the Strong Law of Large Numbers

1973 ◽  
Vol 17 (4) ◽  
pp. 573-581 ◽  
Author(s):  
S. V. Nagaev
Keyword(s):  
Law Of Large Numbers ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Strong Law ◽  
Large Numbers ◽  
Necessary And Sufficient

scholarly journals Weighted Strong Law of Large Numbers for Random Variables Indexed by a Sector

2011 ◽  
Vol 2011 ◽  
pp. 1-16 ◽  
Author(s):  
Przemysław Matuła ◽  
Michał Seweryn
Keyword(s):  
Law Of Large Numbers ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Necessary And Sufficient Conditions ◽  
Independent Random Variables ◽  
Strong Law ◽  
Large Numbers ◽  
Multidimensional Indices ◽  
Necessary And Sufficient

We find necessary and sufficient conditions for the weighted strong law of large numbers for independent random variables with multidimensional indices belonging to some sector.

scholarly journals Complete Moment Convergence for Sung’s Type Weighted Sums ofB-Valued Random Elements

2016 ◽  
Vol 2016 ◽  
pp. 1-8
Author(s):  
Wei Li ◽  
Pingyan Chen ◽  
Soo Hak Sung
Keyword(s):  
Law Of Large Numbers ◽  
Sufficient Conditions ◽  
Weighted Sums ◽  
Complete Moment Convergence ◽  
Real Numbers ◽  
Strong Law ◽  
Moment Convergence ◽  
Large Numbers ◽  
Random Elements ◽  
Necessary And Sufficient

Letp≥1/αand1/2<α≤1.Let{X,Xn,  n≥1}be a sequence of independent and identically distributedB-valued random elements and let{ani,  1≤i≤n,  n≥1}be an array of real numbers satisfying∑i=1naniq=O(n)for someq>p.We give necessary and sufficient conditions for complete moment convergence of the form∑n=1∞n(p-v)α-2E∑i=1naniXi-εnα+v<∞,  ∀ε>0, where0<v<p.A strong law of large numbers for weighted sums of independentB-valued random elements is also obtained.

Two-parameter strong laws and maximal inequalities for U-statistics

1987 ◽  
Vol 107 (1-2) ◽  
pp. 133-151 ◽  
Author(s):  
Terry R. McConnell
Keyword(s):  
Law Of Large Numbers ◽  
Weak Type ◽  
Sufficient Conditions ◽  
Strong Law ◽  
Strong Laws ◽  
Maximal Inequalities ◽  
U Statistics ◽  
Large Numbers ◽  
Necessary And Sufficient ◽  
Two Parameter

SynopsisWe provide necessary and sufficient conditions for two-parameter convergence in the strong law of large numbers for U-statistics. We also obtain weak-type (1,1) inequalities for one and two-sample U-statistics of order 2 which are, in a sense, best possible.

scholarly journals Strong laws of large numbers for the sequence of the maximum of partial sums of i.i.d. random variables

2019 ◽  
Vol 39 (1) ◽  
pp. 19-38
Author(s):  
Shuhua Chang ◽  
Deli Li ◽  
Andrew Rosalsky
Keyword(s):  
Law Of Large Numbers ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Random Variable ◽  
Partial Sums ◽  
Strong Law ◽  
Laws Of Large Numbers ◽  
Strong Laws ◽  
Large Numbers ◽  
Necessary And Sufficient

Let 0 < p ≤ 2, let {Xn; n ≥ 1} be a sequence of independent copies of a real-valued random variable X, and set Sn = X1 + . . . + Xn, n ≥ ­ 1. Motivated by a theorem of Mikosch 1984, this note is devoted to establishing a strong law of large numbers for the sequence {max1≤k≤n |Sk| ; n ≥ ­ 1}. More specifically, necessary and sufficient conditions are given forlimn→∞ max1≤k≤n |Sk|log n−1 = e1/p a.s.,where log x = loge max{e, x}, x ≥­ 0.

Random splittings of an interval

1993 ◽  
Vol 30 (01) ◽  
pp. 131-152
Author(s):  
T. S. Mountford ◽  
S. C. Port
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Law Of Large Numbers ◽  
Unit Interval ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Strong Law ◽  
The Central Limit Theorem ◽  
Large Numbers ◽  
Necessary And Sufficient

Points are independently and uniformly distributed onto the unit interval. The first n—1 points subdivide the interval into n subintervals. For 1 we find a necessary and sufficient condition on {ln } for the events [Xn belongs to the ln th largest subinterval] to occur infinitely often or finitely often with probability 1. We also determine when the weak and strong laws of large numbers hold for the length of the ln th largest subinterval. The strong law of large numbers and the central limit theorem are shown to be valid for the number of times by time n the events [Xr belongs to l r th largest subinterval] occur when these events occur infinitely often.

Random splittings of an interval

1993 ◽  
Vol 30 (1) ◽  
pp. 131-152
Author(s):  
T. S. Mountford ◽  
S. C. Port
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Law Of Large Numbers ◽  
Unit Interval ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Strong Law ◽  
The Central Limit Theorem ◽  
Large Numbers ◽  
Necessary And Sufficient

Points are independently and uniformly distributed onto the unit interval. The first n—1 points subdivide the interval into n subintervals. For 1 we find a necessary and sufficient condition on {ln} for the events [Xn belongs to the ln th largest subinterval] to occur infinitely often or finitely often with probability 1. We also determine when the weak and strong laws of large numbers hold for the length of the ln th largest subinterval. The strong law of large numbers and the central limit theorem are shown to be valid for the number of times by time n the events [Xr belongs to lr th largest subinterval] occur when these events occur infinitely often.

Sign in / Sign up

Export Citation Format

Share Document